The issue turns on a deep divide in measurement theory between what psychometricians call reflective and formative constructs. In a reflective model, an invisible underlying trait—here, a latent quantity of home resources—causes the observable indicators to vary, and the indicators’ mutual correlations determine how much each one counts. In a formative model, the arrows point the other way: the observed components together constitute the composite, and their weights should be justified by what they actually accomplish, such as predicting achievement. Meng and Bolt argue that HRL has always had the hallmarks of a formative index. Books, study supports, parental education, and parental occupation are not interchangeable echoes of a single hidden variable; they are distinct resources that can substitute for one another, and a family can be educationally rich through many different combinations of them.

The dominant practice, however, treats HRL as a latent variable. This tradition brings powerful machinery, including tests of measurement invariance that check whether the indicators relate to the latent construct identically across countries. When those tests fail, researchers typically relax the model through partial invariance, allowing country-specific parameters for the offending items. Wendt and colleagues showed in 2017 that such scaling choices can change estimates of how strongly HRL relates to achievement, and later work on post-conflict societies in the Western Balkans found that the books-at-home item behaved differently across countries, apparently because war and displacement had depleted household libraries independently of other family characteristics. The standard response is to model that anomaly as differential item functioning—a measurement bias to be corrected. Meng and Bolt, drawing on Welzel and Inglehart’s critique and on a 2012 report on socioeconomic status for the US National Assessment of Educational Progress, suggest a provocative alternative: the missing books may not be bias at all, but a genuine resource difference that the index ought to capture.

To test the stakes, the researchers analyzed the 2011 cycle of TIMSS mathematics and PIRLS reading data, covering 185,475 fourth-grade students across 37 countries. Their central tool was a structural equation model in which the five HRL indicators load onto a single latent variable that in turn predicts achievement. If the latent variable faithfully summarizes the achievement-relevant information in its indicators, then each indicator’s factor loading should be roughly proportional to its direct predictive effect on test scores. Any leftover association between an indicator and achievement—detected through modification indices and standardized expected parameter change values—signals a structural violation: the latent variable is over- or under-representing that indicator’s real influence.

The pooled results were striking. Parent-reported number of books in the home carried the strongest loading on the latent HRL variable, about 0.69 for both mathematics and reading, yet its residual effect on achievement was positive and substantial, with expected parameter changes of roughly 0.16 to 0.17. In plain terms, books reported by parents predict achievement considerably more strongly than their loading on the latent variable implies. Parental occupation showed the opposite pattern: despite a healthy loading, its direct relationship with achievement was weaker than the latent model suggested, with negative expected parameter changes around −0.10 to −0.13. The authors offer an intuitive reading. Books in the home reflect not just material resources but a family’s active emphasis on education, whereas parental occupation conveys potential support without saying much about how learning is actually prioritized under the roof.

With the mismatch established, Meng and Bolt compared four ways of building the composite: a latent variable score with weights pooled across countries, a latent variable score with country-specific weights, an achievement-tailored composite with pooled regression weights, and an achievement-tailored composite whose weights come from regressing achievement on the indicators within each country. In the tailored approaches, the regression coefficients themselves become the indicator weights, so the composite is defined by how well its ingredients predict the outcome. Across the full sample, the choice mattered surprisingly little: the various composites correlated at roughly 0.95 or higher, and the average gain in explained achievement variance from tailored weights was only about 0.01 in R-squared. For most countries, the philosophical dispute is nearly academic.

But not everywhere. In a handful of systems, the differences were consequential. In Azerbaijan, the country-specific latent variable and tailored composites correlated at only 0.83, and switching from the latent variable index to the tailored one increased the proportion of explained mathematics achievement variance by about 33 percent. For reading, Norway saw a 44 percent proportional increase in R-squared, from 0.09 to 0.13, when tailored weights replaced country-specific latent variable weights; Ireland gained 22 percent and Saudi Arabia 25 percent. The pattern in Norway and Ireland was the mirror image of good measurement: the latent variable approach, which derives weights purely from how indicators correlate with one another, downweighted the proximal indicators—books and study supports—that actually predict achievement, while upweighting the distal indicators of parental education and occupation. Saudi Arabia displayed its own idiosyncratic profile, with study supports over-weighted and parental education under-weighted by the latent model relative to their true predictive roles.

Perhaps the most counterintuitive finding concerns what happens when researchers do the very thing invariance methodology recommends. Allowing discrimination parameters to vary by country is supposed to improve measurement, yet for many countries it actually reduced the prediction of achievement compared with pooled latent variable weights. The explanation lies in where the weights come from: country-specific loadings are estimated from the internal consistency of the indicators within that country, and there is no guarantee—indeed, no reason to expect—that changes in internal consistency track changes in predictive relevance. A parameter adjustment that makes the measurement model fit better can simultaneously make the composite a worse summary of the achievement-relevant resources a family provides. From a formative perspective, the authors note, partial invariance modeling does not merely fail to fix the problem; it can actively mute meaningful between-country differences by absorbing them into measurement parameters.

The study does not declare a winner. Tailored weights, the authors caution, guarantee better prediction almost by construction, since they are optimized for that purpose—a practice the Cowan report famously warned against, arguing that the relationship between socioeconomic status and achievement should be a finding rather than an optimization exercise. Latent variable weights, by contrast, are defined independently of the outcome and avoid overfitting, but they rest on the assumption that a single hidden cause generates all five indicators, an assumption that recent causal work by VanderWeele shows is frequently violated when indicators have their own direct effects on outcomes. Under such violations, regressing achievement on a latent variable likely underestimates the cumulative influence of the indicators, sometimes substantially.

The broader lesson reaches well beyond the HRL index. International assessments routinely convert background questionnaires into latent constructs—attitudes, motivations, socioeconomic statuses—and then police their comparability with invariance machinery that presumes reflective measurement. Meng and Bolt’s analysis suggests that when a construct is better understood as a bundle of resources than as a hidden trait, that machinery can distort rather than refine cross-country comparisons. The choice between defining equivalence by how indicators relate to each other or by how they relate to an external criterion is not a technical footnote; it changes which countries appear to have strong home-resource effects, by how much, and in which direction. For a field whose rankings move education policy worldwide, the composition of a single index number turns out to be anything but a small detail.

Subject of Research: Methodological comparison of latent variable and achievement-tailored approaches to constructing the Home Resources for Learning index in TIMSS and PIRLS

Article Title: Latent variable or achievement-tailored composite?: The home resources for learning (HRL) index in TIMSS and PIRLS

Article References: Meng, L., & Bolt, D. (2026). Latent variable or achievement-tailored composite?: The home resources for learning (HRL) index in TIMSS and PIRLS. Large-scale Assessments in Education, 14(1), Article 52. https://doi.org/10.1186/s40536-026-00324-8

Image Credits: AI Generated

DOI: 10.1186/s40536-026-00324-8

Keywords: Home Resources for Learning, TIMSS, PIRLS, latent variable models, measurement invariance, formative indicators, reflective indicators, socioeconomic status, structural equation modeling, psychometrics, international large-scale assessments, educational achievement